Black Scholes Merton Option Model Assumptions & Cryptocurrencies
The Famous Black Scholes Options Model
The Black-Scholes-Merton model is a mathematical formula for pricing options contracts. It was developed by Fischer Black, Myron Scholes, and Robert Merton in the early 1970s and has become a standard tool in financial analysis and risk management. (There are plenty of additional resources online that outline the basics of this model). But importantly, this popular, closed form model made it possible to price options using 5 standard inputs:
the Strike of the option
the current Price of underlying
Time to expiration (maturity)
the Risk-Free Rate
Implied Volatility parameter
The Black-Scholes-Merton (BSM) models makes several assumptions about the parameters themselves, some of which do not reflect real world realities. There are also several assumptions that are criticized when applied to traditional markets, but fit characteristics of cryptocurrency markets quite well! Overall, 5 of 7 of these assumptions hold true for cryptocurrency markets:
Implied Volatility is constant - In the real world, the rate of volatility can vary from time period to time period. Volatility in price somewhat mirrors human emotions which vary in intensity at different price levels
The asset follows a random walk in continuous time, the variance of the stock price paths follow a log-normal distribution. - This assumption is typically a critique in traditional markets, as those only trade 5 days a week. However, in cryptocurrency derivative markets, these do trade in continuous time (whether or not they are liquid is another story!) and follow a lognormal distribution (which basically is all values from zero to infinity)
Asset pays no dividends - This is another assumption that fits cryptocurrency markets well (and not all stocks)
The option can only be exercised at expiration (European type option) - Most traditional equity markets allow you to exercise at any point ("American" style); but as of right now, all cryptocurrency option markets are European style
No transaction costs (fees) - Well, this one is a fact of life no matter what market you are trading in. You are going to pay trading fees to someone.
Fractional trading is possible i.e. we can buy/sell 0.x of any given stock - You can't trade fractional sizes of equities or other traditional products, but you can trade any fractional amount of crypto!
Interest rate is known and constant through time - This assumption is very similar to the constant volatility assumption... Interest rates do fluctuate in the market based on supply/demand and economic factors. However, the impact from changes to risk free rates on option delta are significantly less than model pricing changes from implied volatility parameter. We also have written an article for How to find the Risk Free Rate in Python from the US Treasury which could be beneficial to dynamically getting this model parameter.
Tracking Option Activity Via Greeks
Converting option trades to their greeks and tracking overall long or short bias can hint at market participant expectations. We track all BTC and ETH option activity in real time (trading on DeriBit) and calculate associated greeks using the Black Scholes-Merton option pricing model discussed above.
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